Permutations and CombinationsMCQPYQ Dec 22Question 1699 of 251
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If nPr=3024\displaystyle ^nP_r = 3024 and nCr=126\displaystyle ^nC_r = 126, then find n\displaystyle n and r\displaystyle r?

Options

A9, 4
B10, 3
C12, 4
D11, 4
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Correct Answer

✅ Option a — 9, 4

All Options:

  • A9, 4
  • B10, 3
  • C12, 4
  • D11, 4

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Detailed Solution & Explanation

Given the permutation and combination values:
nPr=3024andnCr=126^nP_r = 3024 \quad \text{and} \quad ^nC_r = 126
Recall the relationship between permutations and combinations:
nPr=r!⋅nCr^nP_r = r! \cdot ^nC_r
Substituting the given values into the formula:
3024=r!⋅1263024 = r! \cdot 126
r!=3024126=24r! = \frac{3024}{126} = 24
Since 24=4×3×2×1=4!\displaystyle 24 = 4 \times 3 \times 2 \times 1 = 4!, we have:
r=4r = 4
Now, substitute r=4\displaystyle r=4 into the permutation formula:
nP4=n(n−1)(n−2)(n−3)=3024^nP_4 = n(n-1)(n-2)(n-3) = 3024
We need to express 3024\displaystyle 3024 as a product of 4 consecutive decreasing integers. Let's test values near 30244≈7.4\displaystyle \sqrt[4]{3024} \approx 7.4:
9×8×7×6=30249 \times 8 \times 7 \times 6 = 3024
This matches perfectly, so we have:
n=9n = 9
Thus, n=9\displaystyle n = 9 and r=4\displaystyle r = 4.
Hence, **Option A** is the correct answer.

About This Chapter: Permutations and Combinations

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Factorials, Permutations, Combinations

This chapter deals with the fundamental principles of counting. It covers factorials, circular permutations, restricted permutations, combinations, and the differences between selecting items versus arranging them.

View Official ICAI Syllabus

Exam Strategy Tip

The most common mistake is confusing 'P' (Arrangement) with 'C' (Selection). If order matters (like opening a lock), use P. If order doesn't matter (like choosing a team), use C.

Related Comparison Tables

Permutations vs Combinations: The Quantitative Aptitude Guide

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